- Submitted By: Eryc-Matheka
- Date Submitted: 11/03/2016 2:57 AM
- Category: Business
- Words: 1679
- Page: 7

Distribution of Exceedances

Since Peaks over Threshold (POT) focuses on the realizations exceeding a given (high)

threshold and the threshold method uses data more efficiently, we concentrate on peak over

threshold approach.

The Pickands-Balkema-de Haan Theorem

Suppose that X 1 , X 2 ,..., X n are n independent realisations of a random variable X with a

distribution function F ( x ) . Let xF be the finite or infinite right endpoint of the distribution

F. The distribution function of the excesses over certain (high) threshold u is given by

Fu ( x ) = Pr { X − u ≤ x X > u} =

F ( x + u ) − F (u )

1 − F (u )

for 0 ≤ x < xF − u.

The Pickands-Balkema-de Haan theorem states that if the distribution function F ∈ DA ( H ξ )

then there exists a positive measurable function σ (u ) such that

lim

u→x F

sup { F ( x) − Gξ σ

, (u )

u

}

( x) = 0

0 ≤ x < xF − u

and vice versa, where Gξ ,σ (u ) ( x) denotes the Generalised Pareto distribution (see below).

The above theorem states that as the threshold u becomes large, the distribution of the

excesses over the threshold tends to the Generalised Pareto distribution, provided the

underlying distribution F belongs to the domain of attraction of the Generalised Extreme

Value distribution.

Generalized Pareto Distribution (GPD)

The GPD is a two parameter distribution with distribution function

1

−1

ξ

ξx

1

1

−

+

β

Gξ , β ( x ) =

1 − exp − x

β

(

)

(

)

where β > 0, and where x ≥ 0 when ξ ≥ 0 and 0 ≤ x ≤ − β

ξ ≠0

ξ =0

ξ when ξ 0 then ξ , β is a reparametrized version of the ordinary

Pareto distribution, which has a long history in actuarial mathematics as a model for large

losses; ξ = 0 corresponds to the exponential distribution and ξ < 0 is known as a Pareto type

II distribution.

The first case is the most relevant for risk management purposes since the GPD is heavytailed when ξ > 0. Whereas the normal distribution has moments...